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        <title>API docs for &ldquo;sympy.matrices.matrices.Matrix&rdquo;</title>
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        <body><h1 class="class">Class s.m.m.Matrix(object):</h1><span id="part">Part of <a href="sympy.matrices.matrices.html">sympy.matrices.matrices</a></span><p>known subclasses: <a href="sympy.matrices.matrices.SMatrix.html">sympy.matrices.matrices.SMatrix</a></p><div class="toplevel"><div class="undocumented">Undocumented</div></div><table class="children"><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__init__">__init__</a></td><td><div><p>Matrix can be constructed with values or a rule.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.key2ij">key2ij</a></td><td><div><p>Converts key=(4,6) to 4,6 and ensures the key is correct.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__getattr__">__getattr__</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__getitem__">__getitem__</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__setitem__">__setitem__</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__array__">__array__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.copyin_matrix">copyin_matrix</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.copyin_list">copyin_list</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.hash">hash</a></td><td><div><p>Compute a hash every time, because the matrix elements</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.shape">shape</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__rmul__">__rmul__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.expand 0">expand 0</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.combine">combine</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.subs">subs</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__sub__">__sub__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__mul__">__mul__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__pow__">__pow__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__add__">__add__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__div__">__div__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.multiply">multiply</a></td><td><div><p>Returns self*b</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.add">add</a></td><td><div><p>Returns self+b</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__neg__">__neg__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__eq__">__eq__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__ne__">__ne__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__repr__">__repr__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__str__">__str__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix._format_str">_format_str</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.torepr">torepr</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.tostr">tostr</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.inv">inv</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.__mathml__">__mathml__</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.row">row</a></td><td><div><p>Elementary row operation using functor</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.col">col</a></td><td><div><p>Elementary column operation using functor</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.row_swap">row_swap</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.col_swap">col_swap</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.row_del">row_del</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.col_del">col_del</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">import</span> sympy</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.row_join">row_join</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.col_join">col_join</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.trace">trace</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.submatrix">submatrix</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.slice2bounds">slice2bounds</a></td><td><div><p>Takes slice or number and returns (min,max) for iteration</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.applyfunc">applyfunc</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.reshape">reshape</a></td><td><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *</pre>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.print_nonzero">print_nonzero</a></td><td><div><p>Shows location of non-zero entries for fast shape lookup</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.LUsolve">LUsolve</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.LUdecomposition">LUdecomposition</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.LUdecomposition_Simple">LUdecomposition_Simple</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.LUdecompositionFF">LUdecompositionFF</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.cofactorMatrix">cofactorMatrix</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.minorEntry">minorEntry</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.minorMatrix">minorMatrix</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.cofactor">cofactor</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.jacobian">jacobian</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.QRdecomposition">QRdecomposition</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.simplify">simplify</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.expand">expand</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.cross">cross</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.dot">dot</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.norm">norm</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.normalized">normalized</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.project">project</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.permuteBkwd">permuteBkwd</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.permuteFwd">permuteFwd</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.delRowCol">delRowCol</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.zeronm">zeronm</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.zero">zero</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.eye">eye</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.is_square">is_square</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.is_upper">is_upper</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.is_lower">is_lower</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.is_symbolic">is_symbolic</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.clone">clone</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.det">det</a></td><td><div><p>Compute matrix determinant using Bareis' fraction-free</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.inverse_LU">inverse_LU</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.inverse_GE">inverse_GE</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.rref">rref</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.nullspace">nullspace</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.charpoly">charpoly</a></td><td><span class="undocumented">Undocumented</span></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.berkowitz">berkowitz</a></td><td><div><p>The Berkowitz algorithm.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.berkowitz_det">berkowitz_det</a></td><td><div><p>Computes determinant using Berkowitz method.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.berkowitz_minors">berkowitz_minors</a></td><td><div><p>Computes principal minors using Berkowitz method.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.berkowitz_charpoly">berkowitz_charpoly</a></td><td><div><p>Computes characteristic polynomial minors using Berkowitz method.</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.eigenvals">eigenvals</a></td><td><div><p>Calls polynomials's roots(), doesn't support coeff type right now</p>
</div></td></tr><tr class="function"><td>Function</td><td><a href="#sympy.matrices.matrices.Matrix.eigenvects">eigenvects</a></td><td><span class="undocumented">Undocumented</span></td></tr></table>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__init__">__init__(self, *args):</a></div>
            <div class="functionBody"><div><p>Matrix can be constructed with values or a rule.</p>
<pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>Matrix( (1,2+I), (3,4) ) <span class="py-comment">#doctest:+NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 2 + I]</span>
<span class="py-output">[3,     4]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>Matrix(2, 2, <span class="py-keyword">lambda</span> i,j: (i+1)*j ) <span class="py-comment">#doctest:+NORMALIZE_WHITESPACE</span>
<span class="py-output">[0, 1]</span>
<span class="py-output">[0, 2]</span></pre>
<p>Note: in SymPy we count indices from 0. The rule however counts from 
1.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.key2ij">key2ij(self, key):</a></div>
            <div class="functionBody"><div><p>Converts key=(4,6) to 4,6 and ensures the key is correct.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__getattr__">__getattr__(self, name):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m=Matrix(((1,2+I),(3,4)))
<span class="py-prompt">&gt;&gt;&gt; </span>m  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 2 + I]</span>
<span class="py-output">[3,     4]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.T <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[    1, 3]</span>
<span class="py-output">[2 + I, 4]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.H <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[    1, 3]</span>
<span class="py-output">[2 - I, 4]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__getitem__">__getitem__(self, key):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m=Matrix(((1,2+I),(3,4)))
<span class="py-prompt">&gt;&gt;&gt; </span>m  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 2 + I]</span>
<span class="py-output">[3,     4]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m[1,0]
<span class="py-output">3</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.H[1,0]
<span class="py-output">2 - I</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__setitem__">__setitem__(self, key, value):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m=Matrix(((1,2+I),(3,4)))
<span class="py-prompt">&gt;&gt;&gt; </span>m  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 2 + I]</span>
<span class="py-output">[3,     4]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m[1,0]=9
<span class="py-prompt">&gt;&gt;&gt; </span>m  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 2 + I]</span>
<span class="py-output">[9,     4]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__array__">__array__(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.copyin_matrix">copyin_matrix(self, key, value):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.copyin_list">copyin_list(self, key, value):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.hash">hash(self):</a></div>
            <div class="functionBody"><div><p>Compute a hash every time, because the matrix elements could change.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.shape">shape(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__rmul__">__rmul__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.expand 0">expand 0(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.combine">combine(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.subs">subs(self, a, b):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__sub__">__sub__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__mul__">__mul__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__pow__">__pow__(self, num):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__add__">__add__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__div__">__div__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.multiply">multiply(self, b):</a></div>
            <div class="functionBody"><div><p>Returns self*b</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.add">add(self, b):</a></div>
            <div class="functionBody"><div><p>Returns self+b</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__neg__">__neg__(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__eq__">__eq__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__ne__">__ne__(self, a):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__repr__">__repr__(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__str__">__str__(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix._format_str">_format_str(self, strfunc, rowsep='\n'):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.torepr">torepr(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.tostr">tostr(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.inv">inv(self, method='GE'):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.__mathml__">__mathml__(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.row">row(self, i, f):</a></div>
            <div class="functionBody"><div><p>Elementary row operation using functor</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.col">col(self, j, f):</a></div>
            <div class="functionBody"><div><p>Elementary column operation using functor</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.row_swap">row_swap(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.col_swap">col_swap(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.row_del">row_del(self, i):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.col_del">col_del(self, i):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">import</span> sympy
<span class="py-prompt">&gt;&gt;&gt; </span>M = sympy.matrices.eye(3)
<span class="py-prompt">&gt;&gt;&gt; </span>M.col_del(1)
<span class="py-prompt">&gt;&gt;&gt; </span>M   <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 0]</span>
<span class="py-output">[0, 0]</span>
<span class="py-output">[0, 1]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.row_join">row_join(self, rhs):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.col_join">col_join(self, bott):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.trace">trace(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.submatrix">submatrix(self, keys):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m = Matrix(4,4,<span class="py-keyword">lambda</span> i,j: i+j)
<span class="py-prompt">&gt;&gt;&gt; </span>m   <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[0, 1, 2, 3]</span>
<span class="py-output">[1, 2, 3, 4]</span>
<span class="py-output">[2, 3, 4, 5]</span>
<span class="py-output">[3, 4, 5, 6]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m[0:1, 1]   <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m[0:2, 0:1] <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[0]</span>
<span class="py-output">[1]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m[2:4, 2:4] <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[4, 5]</span>
<span class="py-output">[5, 6]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.slice2bounds">slice2bounds(self, key, defmax):</a></div>
            <div class="functionBody"><div><p>Takes slice or number and returns (min,max) for iteration Takes a 
default maxval to deal with the slice ':' which is (none, none)</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.applyfunc">applyfunc(self, f):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m = Matrix(2,2,<span class="py-keyword">lambda</span> i,j: i*2+j)
<span class="py-prompt">&gt;&gt;&gt; </span>m   <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[0, 1]</span>
<span class="py-output">[2, 3]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.applyfunc(<span class="py-keyword">lambda</span> i: 2*i)  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[0, 2]</span>
<span class="py-output">[4, 6]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.reshape">reshape(self, _rows, _cols):</a></div>
            <div class="functionBody"><div><pre class="py-doctest">
<span class="py-prompt">&gt;&gt;&gt; </span><span class="py-keyword">from</span> sympy <span class="py-keyword">import</span> *
<span class="py-prompt">&gt;&gt;&gt; </span>m = Matrix(2,3,<span class="py-keyword">lambda</span> i,j: 1)
<span class="py-prompt">&gt;&gt;&gt; </span>m   <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 1, 1]</span>
<span class="py-output">[1, 1, 1]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.reshape(1,6)  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 1, 1, 1, 1, 1]</span>
<span class="py-output"></span><span class="py-prompt">&gt;&gt;&gt; </span>m.reshape(3,2)  <span class="py-comment">#doctest: +NORMALIZE_WHITESPACE</span>
<span class="py-output">[1, 1]</span>
<span class="py-output">[1, 1]</span>
<span class="py-output">[1, 1]</span></pre>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.print_nonzero">print_nonzero(self, symb='X'):</a></div>
            <div class="functionBody"><div><p>Shows location of non-zero entries for fast shape lookup &gt;&gt;&gt; 
from sympy import * &gt;&gt;&gt; m = Matrix(2,3,lambda i,j: i*3+j) 
&gt;&gt;&gt; m           #doctest: +NORMALIZE_WHITESPACE [0, 1, 2] [3, 4, 
5] &gt;&gt;&gt; m.print_nonzero()   #doctest: +NORMALIZE_WHITESPACE [ XX] 
[XXX] &gt;&gt;&gt; m = matrices.eye(4) &gt;&gt;&gt; 
m.print_nonzero(&quot;x&quot;)    #doctest: +NORMALIZE_WHITESPACE [x   ] [ 
x  ] [  x ] [   x]</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.LUsolve">LUsolve(self, rhs):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.LUdecomposition">LUdecomposition(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.LUdecomposition_Simple">LUdecomposition_Simple(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.LUdecompositionFF">LUdecompositionFF(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.cofactorMatrix">cofactorMatrix(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.minorEntry">minorEntry(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.minorMatrix">minorMatrix(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.cofactor">cofactor(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.jacobian">jacobian(self, varlist):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.QRdecomposition">QRdecomposition(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.simplify">simplify(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.expand">expand(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.cross">cross(self, b):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.dot">dot(self, b):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.norm">norm(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.normalized">normalized(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.project">project(self, v):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.permuteBkwd">permuteBkwd(self, perm):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.permuteFwd">permuteFwd(self, perm):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.delRowCol">delRowCol(self, i, j):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.zeronm">zeronm(self, n, m):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.zero">zero(self, n):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.eye">eye(self, n):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.is_square">is_square(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.is_upper">is_upper(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.is_lower">is_lower(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.is_symbolic">is_symbolic(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.clone">clone(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.det">det(self):</a></div>
            <div class="functionBody"><div><p>Compute matrix determinant using Bareis' fraction-free algorithm which 
is an extension of the well known Gaussian elimination method. This 
approach is best suited for dense symbolic matrices and will result in a 
determinant with minimal numer of fractions. It means that less term 
rewriting is needed on resulting formulae.</p>
<p>TODO: Implement algorithm for sparse matrices (SFF).</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.inverse_LU">inverse_LU(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.inverse_GE">inverse_GE(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.rref">rref(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.nullspace">nullspace(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.charpoly">charpoly(self, var):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.berkowitz">berkowitz(self):</a></div>
            <div class="functionBody"><pre>The Berkowitz algorithm.

Given N x N matrix with symbolic content, compute efficiently
coefficients of characteristic polynomials of 'self' and all
its square sub-matrices composed by removing both i-th row
and column, without division in the ground domain.

This method is particulary useful for computing determinant,
principal minors and characteristic polynomial, when 'self'
has complicated coefficients eg. polynomials. Semi-direct
usage of this algorithm is also important in computing
efficiently subresultant PRS.

Assuming that M is a square matrix of dimension N x N and
I is N x N identity matrix,  then the following following
definition of characteristic polynomial is begin used:

               charpoly(M) = det(t*I - M)

As a consequence, all polynomials generated by Berkowitz
algorithm are monic.

>>> from sympy import *
>>> x,y,z = symbols('xyz')

>>> M = Matrix([ [x,y,z], [1,0,0], [y,z,x] ])

>>> p, q, r = M.berkowitz()

>>> print p # 1 x 1 M's sub-matrix
(1, -x)

>>> print q # 2 x 2 M's sub-matrix
(1, -x, -y)

>>> print r # 3 x 3 M's sub-matrix
(1, -2*x, -y - y*z + x**2, x*y - z**2)

For more information on the implemented algorithm refer to:

[1] S.J. Berkowitz, On computing the determinant in small
    parallel time using a small number of processors, ACM,
    Information Processing Letters 18, 1984, pp. 147-150

[2] M. Keber, Division-Free computation of subresultants
    using Bezout matrices, Tech. Report MPI-I-2006-1-006,
    Saarbrucken, 2006</pre></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.berkowitz_det">berkowitz_det(self):</a></div>
            <div class="functionBody"><div><p>Computes determinant using Berkowitz method.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.berkowitz_minors">berkowitz_minors(self):</a></div>
            <div class="functionBody"><div><p>Computes principal minors using Berkowitz method.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.berkowitz_charpoly">berkowitz_charpoly(self, x):</a></div>
            <div class="functionBody"><div><p>Computes characteristic polynomial minors using Berkowitz method.</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.eigenvals">eigenvals(self, var=None):</a></div>
            <div class="functionBody"><div><p>Calls polynomials's roots(), doesn't support coeff type right now</p>
</div></div>
            </div>
            <div class="function">
            <div class="functionHeader">def <a name="sympy.matrices.matrices.Matrix.eigenvects">eigenvects(self):</a></div>
            <div class="functionBody"><div class="undocumented">Undocumented</div></div>
            </div></body>
        